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Ultrafilters, congruences, and profinite groups

Rosario Mennuni

Tuesday, 2023-06-13 14:30

It is a well-known fact, with important applications in additive combinatorics and Ramsey theory, that the usual sum of integers may be extended to the space of ultrafilters over Z, yielding a compact right topological semigroup. The analogous construction also goes through for the product.

Recently, B. Šobot introduced two (ternary) notions of congruence on the space above. I will talk about joint work with M. Di Nasso, L. Luperi Baglini, M. Pierobon and M. Ragosta, in which the study of these congruences led us to isolate a class of ultrafilters enjoying characterisations in terms of tensor products, directed sets, profinite groups, and more.